In addition, for the associated Lamé equation (\(x=\operatorname{sn}^2 u=\sin ^2v\) ), the relations [1]}
\(&\begin{array}{l}F\left(-a,a;\frac{1}{2}; \sin ^2v\right)=\cos (2av),\qquad F\left(a,1-a;\frac{3}{2}; \sin ^2v\right)=\frac{\cos [(2a-1)v]}{\cos {v}},\end{array}\nonumber \vspace{5.69054pt}\\&\begin{array}{l}F\left(1-a,a;\frac{3}{2}; \sin ^2v\right)=\frac{\sin [(2a-1)v]}{(2a-1)\sin {v}},\quad F\left(a,2-a;\frac{3}{2}; \sin ^2v\right)=\frac{\sin [(2a-2)v]}{(a-1)\sin (2v)},\end{array}\) 
